Theorem REF  is slightly surprising as the subgroup \(\displaystyle G\)  of \(\displaystyle \mathrm {SL}_{24}\)  generated by \(\displaystyle \mathrm {Spin}_7(\mathbb {C})\)  and \(\displaystyle \mathrm {SL}_3(\mathbb {C})\)  is not \(\displaystyle \mathrm {E}_6\)  (and is even not a subgroup of \(\displaystyle \mathrm {E}_6\) ). We will prove this fact in details in section 3.3, but less us have a glimpse at the argument. Any element of \(\displaystyle \mathrm {J}_{3}(\mathbb {O})\)  can be diagonalized under the action of \(\displaystyle \mathrm {E}_6\) . Hence, any vector of \(\displaystyle \mathrm {J}_{3}(\mathbb {O})\)  can be transformed, under the action of \(\displaystyle \mathrm {E}_6\) , to a vector whose corresponding \(\displaystyle 24*24\)  matrix has rank \(\displaystyle 0,8,16\)  or \(\displaystyle 24\) . On the other hand, the action of \(\displaystyle G\)  on \(\displaystyle \mathrm {J}_{3}(\mathbb {O})\)  preserves the rank. Since there are vectors in \(\displaystyle \mathrm {J}_{3}(\mathbb {O})\)  whose corresponding \(\displaystyle 24*24\)  matrix have rank \(\displaystyle 22\) , we conclude that \(\displaystyle G\)  is not \(\displaystyle \mathrm {E}_6\) . The prehomogeneous space \(\displaystyle (\mathrm {J}_{3}(\mathbb {O}), \mathbb {C}^* \times G)\)  does not appear in the list of Kimura-Sato [1]}. This is not surprising as it is most likely not irreducible as a prohomogeneous space. It thus provides an interesting example of a non-irreducible prehomogeneous vector space which does not split as a sum of lower dimensional prehomogeneous vector spaces.
